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Properties

Label 567.2.ba.a.143.6
Level $567$
Weight $2$
Character 567.143
Analytic conductor $4.528$
Analytic rank $0$
Dimension $132$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(143,567)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("567.143"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([7, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 143.6
Character \(\chi\) \(=\) 567.143
Dual form 567.2.ba.a.341.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.02575 + 1.22245i) q^{2} +(-0.0949069 - 0.538244i) q^{4} +(-1.45366 + 1.21977i) q^{5} +(-2.50057 - 0.864386i) q^{7} +(-2.00866 - 1.15970i) q^{8} -3.02821i q^{10} +(-0.811915 + 0.967602i) q^{11} +(-0.326917 + 0.898197i) q^{13} +(3.62163 - 2.17016i) q^{14} +(4.50524 - 1.63977i) q^{16} +4.01374 q^{17} -7.67461i q^{19} +(0.794495 + 0.666660i) q^{20} +(-0.350017 - 1.98504i) q^{22} +(1.95808 - 5.37979i) q^{23} +(-0.242940 + 1.37778i) q^{25} +(-0.762661 - 1.32097i) q^{26} +(-0.227929 + 1.42795i) q^{28} +(1.22870 + 3.37583i) q^{29} +(-8.76565 + 1.54562i) q^{31} +(-1.03017 + 2.83037i) q^{32} +(-4.11711 + 4.90658i) q^{34} +(4.68933 - 1.79359i) q^{35} +(3.99935 - 6.92708i) q^{37} +(9.38179 + 7.87226i) q^{38} +(4.33449 - 0.764287i) q^{40} +(-3.01647 - 1.09790i) q^{41} +(0.111496 - 0.632326i) q^{43} +(0.597862 + 0.345176i) q^{44} +(4.56799 + 7.91199i) q^{46} +(1.48485 - 8.42098i) q^{47} +(5.50567 + 4.32291i) q^{49} +(-1.43506 - 1.71024i) q^{50} +(0.514475 + 0.0907159i) q^{52} +(-11.3329 - 6.54305i) q^{53} -2.39691i q^{55} +(4.02037 + 4.63618i) q^{56} +(-5.38712 - 1.96075i) q^{58} +(2.02915 + 0.738551i) q^{59} +(-3.03575 - 0.535284i) q^{61} +(7.10196 - 12.3010i) q^{62} +(2.39111 + 4.14152i) q^{64} +(-0.620364 - 1.70444i) q^{65} +(1.63854 - 1.37490i) q^{67} +(-0.380931 - 2.16037i) q^{68} +(-2.61754 + 7.57223i) q^{70} +(0.154696 - 0.0893139i) q^{71} +(9.96274 - 5.75199i) q^{73} +(4.36563 + 11.9945i) q^{74} +(-4.13081 + 0.728373i) q^{76} +(2.86663 - 1.71775i) q^{77} +(-0.935206 - 0.784731i) q^{79} +(-4.54896 + 7.87903i) q^{80} +(4.43628 - 2.56129i) q^{82} +(-5.11532 + 1.86183i) q^{83} +(-5.83462 + 4.89582i) q^{85} +(0.658617 + 0.784910i) q^{86} +(2.75300 - 1.00201i) q^{88} -5.63916 q^{89} +(1.59387 - 1.96342i) q^{91} +(-3.08147 - 0.543347i) q^{92} +(8.77111 + 10.4530i) q^{94} +(9.36123 + 11.1563i) q^{95} +(-5.97585 - 1.05370i) q^{97} +(-10.9320 + 2.29615i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.02575 + 1.22245i −0.725318 + 0.864400i −0.995136 − 0.0985112i \(-0.968592\pi\)
0.269818 + 0.962911i \(0.413036\pi\)
\(3\) 0 0
\(4\) −0.0949069 − 0.538244i −0.0474534 − 0.269122i
\(5\) −1.45366 + 1.21977i −0.650097 + 0.545497i −0.907101 − 0.420914i \(-0.861709\pi\)
0.257003 + 0.966411i \(0.417265\pi\)
\(6\) 0 0
\(7\) −2.50057 − 0.864386i −0.945126 − 0.326707i
\(8\) −2.00866 − 1.15970i −0.710170 − 0.410017i
\(9\) 0 0
\(10\) − 3.02821i − 0.957603i
\(11\) −0.811915 + 0.967602i −0.244802 + 0.291743i −0.874428 − 0.485155i \(-0.838763\pi\)
0.629627 + 0.776898i \(0.283208\pi\)
\(12\) 0 0
\(13\) −0.326917 + 0.898197i −0.0906704 + 0.249115i −0.976736 − 0.214445i \(-0.931206\pi\)
0.886066 + 0.463560i \(0.153428\pi\)
\(14\) 3.62163 − 2.17016i 0.967922 − 0.580000i
\(15\) 0 0
\(16\) 4.50524 − 1.63977i 1.12631 − 0.409944i
\(17\) 4.01374 0.973474 0.486737 − 0.873549i \(-0.338187\pi\)
0.486737 + 0.873549i \(0.338187\pi\)
\(18\) 0 0
\(19\) − 7.67461i − 1.76068i −0.474348 − 0.880338i \(-0.657316\pi\)
0.474348 − 0.880338i \(-0.342684\pi\)
\(20\) 0.794495 + 0.666660i 0.177654 + 0.149070i
\(21\) 0 0
\(22\) −0.350017 − 1.98504i −0.0746239 − 0.423213i
\(23\) 1.95808 − 5.37979i 0.408288 − 1.12176i −0.549801 − 0.835295i \(-0.685297\pi\)
0.958090 − 0.286468i \(-0.0924812\pi\)
\(24\) 0 0
\(25\) −0.242940 + 1.37778i −0.0485879 + 0.275556i
\(26\) −0.762661 − 1.32097i −0.149570 − 0.259063i
\(27\) 0 0
\(28\) −0.227929 + 1.42795i −0.0430745 + 0.269857i
\(29\) 1.22870 + 3.37583i 0.228164 + 0.626876i 0.999960 − 0.00899503i \(-0.00286324\pi\)
−0.771795 + 0.635871i \(0.780641\pi\)
\(30\) 0 0
\(31\) −8.76565 + 1.54562i −1.57436 + 0.277602i −0.891524 − 0.452973i \(-0.850363\pi\)
−0.682833 + 0.730575i \(0.739252\pi\)
\(32\) −1.03017 + 2.83037i −0.182110 + 0.500343i
\(33\) 0 0
\(34\) −4.11711 + 4.90658i −0.706078 + 0.841471i
\(35\) 4.68933 − 1.79359i 0.792641 − 0.303171i
\(36\) 0 0
\(37\) 3.99935 − 6.92708i 0.657489 − 1.13880i −0.323774 − 0.946134i \(-0.604952\pi\)
0.981264 − 0.192670i \(-0.0617148\pi\)
\(38\) 9.38179 + 7.87226i 1.52193 + 1.27705i
\(39\) 0 0
\(40\) 4.33449 − 0.764287i 0.685343 − 0.120844i
\(41\) −3.01647 − 1.09790i −0.471093 − 0.171464i 0.0955544 − 0.995424i \(-0.469538\pi\)
−0.566647 + 0.823960i \(0.691760\pi\)
\(42\) 0 0
\(43\) 0.111496 − 0.632326i 0.0170030 − 0.0964289i −0.975125 − 0.221654i \(-0.928854\pi\)
0.992128 + 0.125226i \(0.0399654\pi\)
\(44\) 0.597862 + 0.345176i 0.0901311 + 0.0520372i
\(45\) 0 0
\(46\) 4.56799 + 7.91199i 0.673513 + 1.16656i
\(47\) 1.48485 − 8.42098i 0.216587 − 1.22833i −0.661544 − 0.749906i \(-0.730099\pi\)
0.878131 − 0.478420i \(-0.158790\pi\)
\(48\) 0 0
\(49\) 5.50567 + 4.32291i 0.786525 + 0.617558i
\(50\) −1.43506 − 1.71024i −0.202949 − 0.241865i
\(51\) 0 0
\(52\) 0.514475 + 0.0907159i 0.0713449 + 0.0125800i
\(53\) −11.3329 − 6.54305i −1.55669 − 0.898757i −0.997570 − 0.0696726i \(-0.977805\pi\)
−0.559123 − 0.829085i \(-0.688862\pi\)
\(54\) 0 0
\(55\) − 2.39691i − 0.323200i
\(56\) 4.02037 + 4.63618i 0.537245 + 0.619535i
\(57\) 0 0
\(58\) −5.38712 − 1.96075i −0.707363 − 0.257459i
\(59\) 2.02915 + 0.738551i 0.264173 + 0.0961512i 0.470711 − 0.882287i \(-0.343997\pi\)
−0.206538 + 0.978439i \(0.566220\pi\)
\(60\) 0 0
\(61\) −3.03575 − 0.535284i −0.388688 − 0.0685361i −0.0241106 − 0.999709i \(-0.507675\pi\)
−0.364577 + 0.931173i \(0.618786\pi\)
\(62\) 7.10196 − 12.3010i 0.901950 − 1.56222i
\(63\) 0 0
\(64\) 2.39111 + 4.14152i 0.298889 + 0.517690i
\(65\) −0.620364 − 1.70444i −0.0769467 − 0.211409i
\(66\) 0 0
\(67\) 1.63854 − 1.37490i 0.200180 − 0.167971i −0.537187 − 0.843463i \(-0.680513\pi\)
0.737367 + 0.675492i \(0.236069\pi\)
\(68\) −0.380931 − 2.16037i −0.0461947 − 0.261983i
\(69\) 0 0
\(70\) −2.61754 + 7.57223i −0.312856 + 0.905055i
\(71\) 0.154696 − 0.0893139i 0.0183591 − 0.0105996i −0.490792 − 0.871277i \(-0.663293\pi\)
0.509151 + 0.860677i \(0.329959\pi\)
\(72\) 0 0
\(73\) 9.96274 − 5.75199i 1.16605 − 0.673220i 0.213304 − 0.976986i \(-0.431578\pi\)
0.952747 + 0.303766i \(0.0982442\pi\)
\(74\) 4.36563 + 11.9945i 0.507494 + 1.39433i
\(75\) 0 0
\(76\) −4.13081 + 0.728373i −0.473836 + 0.0835501i
\(77\) 2.86663 − 1.71775i 0.326683 − 0.195756i
\(78\) 0 0
\(79\) −0.935206 − 0.784731i −0.105219 − 0.0882891i 0.588661 − 0.808380i \(-0.299655\pi\)
−0.693879 + 0.720091i \(0.744100\pi\)
\(80\) −4.54896 + 7.87903i −0.508589 + 0.880902i
\(81\) 0 0
\(82\) 4.43628 − 2.56129i 0.489906 − 0.282847i
\(83\) −5.11532 + 1.86183i −0.561480 + 0.204362i −0.607140 − 0.794595i \(-0.707683\pi\)
0.0456599 + 0.998957i \(0.485461\pi\)
\(84\) 0 0
\(85\) −5.83462 + 4.89582i −0.632853 + 0.531027i
\(86\) 0.658617 + 0.784910i 0.0710205 + 0.0846390i
\(87\) 0 0
\(88\) 2.75300 − 1.00201i 0.293470 − 0.106814i
\(89\) −5.63916 −0.597750 −0.298875 − 0.954292i \(-0.596611\pi\)
−0.298875 + 0.954292i \(0.596611\pi\)
\(90\) 0 0
\(91\) 1.59387 − 1.96342i 0.167083 − 0.205822i
\(92\) −3.08147 − 0.543347i −0.321266 − 0.0566478i
\(93\) 0 0
\(94\) 8.77111 + 10.4530i 0.904671 + 1.07814i
\(95\) 9.36123 + 11.1563i 0.960442 + 1.14461i
\(96\) 0 0
\(97\) −5.97585 − 1.05370i −0.606755 − 0.106987i −0.138174 − 0.990408i \(-0.544123\pi\)
−0.468581 + 0.883421i \(0.655235\pi\)
\(98\) −10.9320 + 2.29615i −1.10430 + 0.231946i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 567.2.ba.a.143.6 132
3.2 odd 2 189.2.ba.a.101.17 ✓ 132
7.5 odd 6 567.2.bd.a.467.17 132
21.5 even 6 189.2.bd.a.47.6 yes 132
27.4 even 9 189.2.bd.a.185.6 yes 132
27.23 odd 18 567.2.bd.a.17.17 132
189.131 even 18 inner 567.2.ba.a.341.6 132
189.166 odd 18 189.2.ba.a.131.17 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.17 ✓ 132 3.2 odd 2
189.2.ba.a.131.17 yes 132 189.166 odd 18
189.2.bd.a.47.6 yes 132 21.5 even 6
189.2.bd.a.185.6 yes 132 27.4 even 9
567.2.ba.a.143.6 132 1.1 even 1 trivial
567.2.ba.a.341.6 132 189.131 even 18 inner
567.2.bd.a.17.17 132 27.23 odd 18
567.2.bd.a.467.17 132 7.5 odd 6